1. Field of the Invention
The present invention on is directed to a method for designing a selective RF pulse for use in a magnetic resonance apparatus, as well as to a selective RF pulse designed in accordance with the method.
2. Description of the Prior Art
Magnetic resonance is a known technology for, among other things, acquiring images of the inside of the body of an examination subject. In a magnetic resonance apparatus, rapidly switched gradient fields that are generated by a gradient coil system are thereby superimposed on a static basic magnetic field that is generated by a basic field magnet. The magnetic resonance apparatus also has a radiofrequency (RF) system that emits radiofrequency energy into the examination subject for triggering magnetic resonance signals and picks up the magnetic resonance signals, on the basis of which magnetic resonance images are produced.
When the examination subject is thereby exposed to the static, homogeneous basic magnetic field, those atomic nuclei of the examination subject having a magnetic moment have a resonant frequency that is directly proportional to the strength of the basic magnetic field. If the atomic nuclei of a prescribable isotope that is bonded in a prescribable chemical bond, for example 1H in H2O, were excited with an RF pulse having the same frequency as the atomic nuclei bonded in this way, then all of these atomic nuclei would exhibit identical resonance and emit undifferentiated magnetic resonance signals that would contain no spatial information as to the distribution of the atomic nuclei in the examination subject.
For a spatially specific magnetic resonance signal, one standard method is to superimpose a magnetic gradient field on the static basic magnetic field during the excitation with RF pulses. As a result, the atomic nuclei experience different magnetic field strengths at different locations along the gradient of the gradient field and therefore exhibit resonance at different frequencies. A slice without any thickness whatsoever would be excited with a “monochromatic” RF pulse would have only one of the resonant frequencies. A thin, three-dimensional cuboid, however, is desired as a slice, so that the exciting RF pulse must have a specific bandwidth of neighboring frequencies around its center frequency so that it can excite the desired, narrow spatial region of the slice thickness along the gradient.
Due to non-linearities of Bloch's equations, the design problem for selective RF pulses generally also is not of a linear nature. One possible solution of this problem thereby makes use of an algorithm known as the Shinnar-LeRoux algorithm that is described in greater detail in the article by J. Pauly at el., “parameter Relations for the Shinnar-LeRoux Selective Excitation Pulse Design Algorithm”, IEEE Transactions on Medical Imaging, Vol. 10, No.1. March 1991, pages 56 through 65. In accord therewith, there is a definitive relationship between an RF pulse B1(t) and two polynomials An(z) and Bn(z) via the Shinnar-LeRoux transformation: with t representing time and z being a complex variable. A solution of the design problem for a selective RF pulse proceeds from the fact that the polynomial Bn(eiγGxΔt) is proportional to the sine of half the flip angle at the location x, i.e. to sin(α(x)/2), given a flip angle distribution α(x) that is prescribed for a selection gradient direction. The definition of the complex variable z as z=eiγGΔt effects a presentation of the polynomial Bn(z) on the circle with radius 1, whereby γ is the gyromagnetic ratio, G is the selection gradient amplitude and Δt is the duration of a section of the imagined RF pulse divided into many constant sections. First, that polynomial Bn(z) that optimally approaches the ideal slice profile is determined, making use, for example, of the Parks-McClellan algorithm. Subsequently, An(z) is determined in agreement with Bn(z) with the additional condition that the resulting RF pulse has minimum energy, to which end the polynomial An(z) is selected with phase minimization and determined from Bn(z) via the Hilbert transformation. After An(z) and Bn(z) have been determined, the RF pulse is determined by means of the inverse Shinnar-LeRoux transformation.
Further, the polynomials An(z) and Bn(z) are linked to one another via the equation An(z)·An※(z)+Bn(z)·Bn※(z)=1, resulting in the magnitude |An(z)| of the polynomial An(z) for cos(α(x)/2) and the phase thereof being derived as an unambiguous function of |An(z)|, and thus of |Bn(z)| as a consequence of the aforementioned phase minimization. Given excitation of a steady state magnetization with an RF pulse designed in this way, the transverse magnetization then corresponds to twice the convolution product of the two polynomials An(z) [sic] and Bn(z). As is known, the magnitude of the convolution product is 2 sin(α(x)/2)cos(α(x)/2)=sin(α(x)). The phase distribution of the transverse magnetization along the selection gradient direction x—referred to in short as the azimuth phase distribution—thereby arises from the phase sum of An※(z) and Bn(z), i.e. from the phase difference of An(z) and Bn(z).